Justifications-on-demand as a device to promote shifts of attention associated with relational thinking in elementary arithmetic

Student responses to arithmetical questions that can be solved by using arithmetical structure can serve to reveal the extent and nature of relational, as opposed to computational thinking. Here, student responses to probes which require them to justify-on-demand are analysed using a conceptual framework which highlights distinctions between different forms of attention. We analyse a number of actions observed in students in terms of forms of attention and shifts between them: in the short-term (in the moment), medium-term (over several tasks), and long-term (over a year). The main factors conditioning students´ attention and its movement are identified and some didactical consequences are proposed

3D printed microchannels for sub-nL NMR spectroscopy

Nuclear magnetic resonance (NMR) experiments on subnanoliter (sub-nL) volumes are hindered by the limited sensitivity of the detector and the difficulties in positioning and holding such small samples in proximity of the detector. In this work, we report on NMR experiments on liquid and biological entities immersed in liquids having volumes down to 100 pL. These measurements are enabled by the fabrication of high spatial resolution 3D printed microfluidic structures, specifically conceived to guide and confine sub-nL samples in the sub-nL most sensitive volume of a single-chip integrated NMR probe. The microfluidic structures are fabricated using a two-photon polymerization 3D printing technique having a resolution better than 1 \u3bcm3. The high spatial resolution 3D printing approach adopted here allows to rapidly fabricate complex microfluidic structures tailored to position, hold, and feed biological samples, with a design that maximizes the NMR signals amplitude and minimizes the static magnetic field inhomogeneities. The layer separating the sample from the microcoil, crucial to exploit the volume of maximum sensitivity of the detector, has a thickness of 10 \u3bcm. To demonstrate the potential of this approach, we report NMR experiments on sub-nL intact biological entities in liquid media, specifically ova of the tardigrade Richtersius coronifer and sections of Caenorhabditis elegans nematodes. We show a sensitivity of 2.5x1013spins/ Hz1/2on1H nuclei at 7 T, sufficient to detect 6 pmol of1H nuclei of endogenous compounds in active volumes down to 100 pL and in a measurement time of 3 hours. Spectral resolutions of 0.01 ppm in liquid samples and of 0.1 ppm in the investigated biological entities are also demonstrated. The obtained results may indicate a route for NMR studies at the single unit level of important biological entities having sub-nL volumes, such as living microscopic organisms and eggs of several mammalians, humans included